HBT MODELING

The UCSD High Speed Devices Group, in collaboration with the HBT Model Working Group (including Rockwell, TRW, Hewlett-Packard, Texas Instruments, Cadence, Silvaco, Meta-Software, University of Illinois) has been working to develop better SPICE models for heterojunction bipolar transistors (HBTs). This effort has been partially supported by ARPA under the High Speed Circuit Design Program and monitored by NCCOSC RDT&E Div. (NRaD). The following is a summary of the HBT equations.

1.0 Introduction
2.0 Model Topology
3.0 Highlights of Changes in Equations from BJT Model
4.0 Model Equations
5.0 Summary of Model Parameters
6.0 Figures
7.0 Appendix A
8.0 Appendix B
9.0 Goodies

1.0 Introduction

This document summarizes the HBT model equations formulated for the ARPA High Speed Circuit Design Program. They represent a consolidation of inputs from the HBT Model Working Group, as well as from a variety of other contributors. Specific major contributions have been provided by L. Camnitz of HP and C.Grossman of TRW. There has also been a significant input from the group developing the VBIC95 BJT model (under the auspices of BCTM), particularly by Dr. Jerry Seitchik of Texas Instruments.

This document provides, first, a description of highlights of the changes in equations from those of the BJT model. Following this, a listing of the full set of equations of the model is given, and finally, a summary of model parameters.

In formulating the present model, backward compatibility with the "standard" SPICE Gummel-Poon model has been sacrificed. Thus while most formulations from Berkeley SPICE have been grandfathered, not all are retained.

The HBT model allows various degrees of trade-off between accuracy and computational complexity. Flags permit turning off several features of the model in order to allow faster computation or easier convergence. Models of lower complexity are obtained by deleting some of the nodes and corresponding equations.

2.0 Model Topology

An npn transistor is assumed. The equivalent circuit network for the large-signal HBT model, is shown in fig. 1. There are up to 5 external nodes (E,B,C,Th,S) and up to 7 internal nodes (Ei, Bi, Ci, Bx, Cx, Ex, T ). If the flag SELFT is set "false", the temperature nodes Th and T are not defined.

For ac analysis, a corresponding small-signal model is defined. The topology of this model is shown in fig.2. It is noteworthy that the model contains dependent current sources that have the function of transcapacitances as well as transconductances.

3.0 Highlights of Changes in Equations from BJT Model

1) Self-heating

A thermal subcircuit is established, based on the equivalence I<-> Power and V<-> Temperature. The current (power) source is the electrical power dissipation associated with I*V for the non-energy storage elements of the electrical model. The transistor has a single thermal resistance and thermal capacitance connecting it to an external temperature node. The external node assignment follows the suggestion of C. Grossman and parallels power transistor formulations. An interested user can introduce a more complex thermal circuit by using the external temperature node with external elements in the SPICE file. The external node, Th, can also be used to represent an external heat-sink, or to represent thermal interactions between devices. The equations of self-heating are implicit in the topology of fig.1, and approximately correspond to:

P=IC*VCE + IB * VBE

CTH * d delT/dt = P - delT/RTH

Here delT represents temperature rise of device above temperature of external Th node. To avoid computational complexity, the self-consistent (or "dynamic") temperature Td is not used to calculate every one of the transistor parameters; it is used for collector and base currents, beta values and forward transit time. Remaining parameter dependences are computed with the operating temperature, Top, which is an estimated temperature of the device in operation, specified by the user as an instance parameter or global parameter in a conventional fashion. To minimize computational complexity and optimize convergence, the user may turn off the self-consistent temperature calculation (by setting the flag SELFT="false"). Then the device temperature is taken to be Top for the evaluation of all variables.

2) Collector current vs bias conditions

The standard Gummel-Poon relation relating Ic to Vbe and Vbc is replaced by a more complex equation that better describes HBTs. The equation allows for an ideality factor that varies with bias (as found in many HBTs having a potential barrier at the B-E junction). It also allows for effects associated with potential barriers at the B-C junction (which can occur in DHBTs, with wide-bandgap collectors).

IC = IS * [ exp(qVBE/NF/KT) - exp(qVBC/NR/KT) ] * [qb + IS* exp(qVBE/NA/KT) / ISA + IS*exp(qVBC/NB/KT)/ISB]^ -1

qb remains in approximately the form of the standard BJT model.

NA and NB are new parameters, independent of temperature.

ISA and ISB are new parameters, whose temperature dependence follows the relation

IS/ISA = K1 * exp( Ea/KT)

IS/ISB = K2 * exp( Eb/KT)

Here the activation energies Ea and Eb are physically related to the corresponding junction barrier heights.

3) Emitter-Base Parasitic Diode

An extra diode between B' and E' nodes is allowed, with a resistance REX in series. This allows for experimentally observed base current components which saturate at high bias. The equations are implicit in the topology.

4) Partitioning of base and collector resistances into extrinsic and intrinsic portions

As evident in the topology, the resistances are made to correspond more clearly to the extrinsic and intrinsic parts of the transistor. Inclusion of the two resistances is important to formulate high speed HBT characteristics accurately. The capacitances and parasitic diodes from base to collector are allowed to differ between extrinsic and intrinsic base regions.

5) B-C Capacitance vs bias conditions

The intrinsic base-collector capacitance often varies with VBC in a complex fashion due to the fact that the depletion region reaches through to the n+ layer at some voltages, not at others. Capacitance varies with collector current since the depletion region charge is affected by the mobile electron density. The decrease of Cbc at the critical collector current ICRIT is accounted for. Typically ICRIT is the collector current for which the traveling electron charge is equal to the collector doping. The capacitance is CCMIN for fully depleted collector.

6) Avalanche breakdown of B-C junction

A current source is established between Ci and Bi nodes, with current IBK given by the voltage-dependent multiplication factor. This factor is computed using the well used expression:

Mfactor = 1 / (1 - (Vcb/BVC)^NBC)

If Vbci is near to the breakdown voltage, then the multiplication factor is computed with a linear extrapolation formula that avoids the singularity of the expression used. A flag (BKDN) allows turning the avalanche contribution on or off, to potentially alleviate convergence problems.

7) Charge conservation

In order to maintain accuracy of SPICE circuit computations, it is important to write equations for charge storage at junctions that are explicit functions of bias currents and voltages (as opposed to being specified as an integral of capacitance over voltage, relying on numerical integration to determine the charges stored).

In the HBT model, a formulation developed by Stretch Camnitz is used. This formulation includes effects of how base-emitter charge Qbe depends on Vbc as well as Vbe; similarly, the dependence of Qbc on Ic (including Kirk effect, etc) is taken into account. To formulate these effects consistently, it is necessary to include transcapacitances in the small signal model, that is, elements for which the current is proportional to the time-derivative of a voltage other than the capacitor terminal voltage (for example, Ibc has a contribution of the form C1*dVbe/dt, in addition to a contribution following C2*dVbc/dt, etc.).

8) TF vs bias conditions

TF is adjusted to allow for increase with applied voltage, with bias dependent depletion region width, and to allow for base pushout. The treatment of base transit time and collector depletion region transit time are largely decoupled.

The formalism for the treatment of base-collector depletion region charge follows the derivation by L. Camnitz, taking quite accurate account of the voltage and current dependences of the stored charge.

The behavior of base transit time with bias conditions is also addressed. The standard Gummel-Poon formalism that couples the expressions for base charge used for dc Icf calculation and those used for ac calculations is not used, however. The Ikf parameters have been eliminated from the ac model. The Early voltage is used to modulate the base charge, and additionally, contributions to the base charge from conduction band "notches" or valleys appearing at the emitter-base and collector-base heterojunctions are included.

9) Excess phase formulation

The equation formulation includes transcapacitors, which can be used to quantify the excess phase characteristics of the transistor. Such a formulation is close to what has been discussed by J.Fossum, and to what is implemented in the MEXTRAM model.

10) TR vs bias

Charge storage in reverse operation in HBTs can be associated with both intrinsic and extrinsic portions of the base-collector junction. Both are provided for in this model. In the extrinsic junction region, a separate time constant TRX is used.

11) Temperature dependences of HBT Parameters

Power law dependences of parameters on temperature are specified in most cases. We distinguish between using the self-consistent temperature to evaluate parameters (Tdev), and using a "good temperature guess" (Top) provided by the user as an instance parameter to evaluate the parameters. Different sets of currents and charges are evaluated with Tdev and Top. Evaluation with Top saves computational time.

4.0 Model Equations

Branch voltages:

Definitions used in equations below:

Vbei= V(Bi)-V(Ei)

Vbci= V(Bi)-V(Ci)

Vbcx= V(Bi)-V(Cx)

Vcs= V(Cx)-V(S)

Branch currents:

Definitions used in equations below:

Icc: From Ci to Ei

Ibei: From Bi to Ei

Ibex: From Ep to Ei

Ibci: From Bi to Ci

Ibcx: From Bx to Cx

Ith: From Tcom to T

Current Flow Contributions

Intrinsic collector current contributions

The electron flow between Ei and Ci nodes is computed in accord with the Gummel-Poon model, with modifications to take into account the potential spike that can appear at the base-emitter or base-collector junctions of HBTs. It is convenient to separate the electron current into forward and reverse components, Icf and Icr.

Icf = IS * [exp(qVbei/NF/KT) - 1] / D

Icr = IS * [exp(qVbci/NR/KT) -1] / D

Here D= qb + IS* exp(qVbei/NA/KT) / ISA + IS*exp(qVbci/NB/KT)/ISB

ISA, ISB, NA and NB are new parameters. ISA and ISB are roughly indicative of the transition currents from base transport controlled to potential barrier controlled current flow.

qb partially retains the form of the standard BJT model (representing fractional increase in the base charge associated with bias changes).

qb= q1/2 * [1+(1+4*q2)^0.5]

q1=1/ [ 1 - Vbci/VAF -Vbei/VAR]

q2=IS/IK*[exp(qVbei/NF/KT)-1]

qb omits the reverse knee current contribution. As noted below, qb is not used to define the ac model in the fashion of the Gummel-Poon model.

The self-consistent or "dynamic"device temperature Td is used to compute IS, ISA, ISB, KT in the above equations. NF, NA and NB are taken to be temperature independent.

The total collector current Icc is given by

Icc=Icf - Icr

The formulation uses the parameters IS, NF, VAF, VAR, IK established in the SPICE BJT model, in addition to ISA, ISB, NA, and NB described above.

Intrinsic Base-Emitter Diode

Ideal and non-ideal components are included:

Ibei= Icf / BF + ISE * [exp (q Vbei/ NE /KT) - 1]

The parameters are the well-known quantities from the BJT model BF, ISE and NE. BF and ISE are temperature dependent, evaluated with Td , the self-consistent temperature (as is KT above). NE is similarly allowed to be temperature dependent (evaluated with the self-consistent temperature Td).

Extrinsic Base-Emitter Diode

The model includes a (new) diode connected between Ex and E nodes, which together with an associated series resistance Rex described below may be used to model contributions from emitter edges.

Ibex = ISEX * [exp (q Vbex/ NEX /KT) - 1]

New parameters are introduced, ISEX, and NEX. These parameters and KT above are computed with the self-consistent temperature Td.

Intrinsic Base-Collector Diode

Ideal and non-ideal components are included:

Ibci = Icr / BR + ISC * [ exp (q Vbci/ NC /KT) - 1]

The parameters used are BR, ISC, and NC. Top is used to evaluate the temperature dependence.

Intrinsic Base-Collector Breakdown Current

Ibk is current between collector and base nodes generated due to avalanche breakdown of base-collector junction. If the user-specified parameter BKDN is "true", then Ibk is determined according to:

Ibk= (Mf -1) * Icf

(Otherwise, Ibk=0). Here Mf is the multiplication factor associated with the BC junction at the given voltage, and Icf is the forward electron current (as computed above in the absence of multiplication). Mf is calculated with a physically based expression, modified to avoid the singularity at Vbci=-BVC. Mf depends exclusively on the intrinsic base-intrinsic collector voltage Vbci. If -Vbci closely approaches or exceeds BVC (-Vbci>FA*BVC, with FA typically chosen to be 0.95), then the multiplication factor is computed according to a constant slope expression.

Mf = 1 / [ 1- (-Vbci/BVC)^NBC ] for KTop/q< -Vbci < FA*BVC

Mf = 1 for -Vbci > KTop/q

Mf = Mfl + gl *(-Vbci-FA*BVC) for -Vbci > FA*BVC

Here Mfl and gl are the values of Mf and its derivative with respect to voltage, evaluated at the voltage -Vbci=FA*BVC:

Mfl=1 / (1-FA^NBC)

gl = Mfl*(Mfl-1)*NBC/(FA*BVC)

Parameters required: BVC, NBC, FA, BKDN. Temperature dependences are evaluated with Top.

Extrinsic Base-Collector Diode

This diode has customary I-V characteristics, with its own saturation current and ideality factor.

Ibcx= ISCX * [ exp(q Vbcx/ NCX /KTop) - 1]

New parameters are ISCX and NCX. Top describes the temperature dependence.

Substrate-Extrinsic Collector Diode

This diode provides allowances for conducting substrates, primarily of interest for SiGe HBTs.

Ics= ICS * [ exp(-q Vcs/ NCS /KTop) - 1]

New parameters introduced are ICS and NCS. Top is used for the temperature dependence.

In accordance with the model topology, the external currents through the nodes E,B and C are:

Ib= Ibei + Ibex - Ibk + Ibci + Ibcx

Ic= Icc + Ibk -Ibci-Ibcx - Ics

Parasitic Resistances

Emitter resistance:

RE (with no voltage or current dependences) is connected between nodes Ei and E.

An additional resistance, REX (with no voltage or current dependences) is connected between Bx and Ex nodes.

The associated parameters are: RE and REX. Temperature dependences are determined through Top (best guess operating temperature).

Base resistance:

(1) RBX (with no voltage or current dependences) is connected between nodes B and Bx.

(2) RBI is connected between nodes Bx and Bi. In BJT SPICE RBI is allowed to be voltage and current dependent, but we omit this for simplicity (since it is not relevant to HBTs).

Parameters for base resistance: RBI and RBX. Temperature dependence determined thorugh Top.

Collector resistance:

(1) RCX (with no voltage or current dependences), connected between nodes C and Cx.

(2) RCI, connected between nodes Cx and Ci. Provisions will be made in a future extension of the model, for RCI to vary with voltage and current, as the depletion region width changes, but at present RCI is bias independent. Top describes the temperature dependence.

Charge Storage

Introduction

The formulation of HBT charge storage has many novel features, most of which were developed by Dr. L. Camnitz of Hewlett-Packard. In this formulation, expressions are developed for the charge stored in the junction regions as a function of collector current and junction voltages. Capacitances and storage times can be calculated from the resulting expressions. The treatment of the two is thus unified, such that junction capacitances and storage times are not independently specified, but emerge together when proper account is taken of the current and voltage dependences of the charge stored. The formulation also leads to the appearance of transcapacitances in the ac model: the charge associated with one junction is in general a function of the voltage developed across other junctions.

The partitioning of the charge associated with current flow between the base-emitter and base-collector junctions can be varied in the model specification, in order to allow variation of the excess phase characteristics of devices.

The charge associated with depletion regions is calculated according to a model that is more comprehensive than that of standard SPICE. It takes into account the fact that the capacitance frequently limits at some minimum value, reached when lightly doped layers are fully depleted.

Base-Emitter Charge

The overall charge stored at the base-emitter junction has components associated with the base-emitter depletion layer, Qbej (taken to be current independent), as well as collector current-dependent charge, Qbediff, which corresponds to a portion of the base charge, and the (collector-current dependent) base-collector charge.

Qbe= Qbej + Qbediff

Base-Emitter Depletion Charge Qbej::

The depletion charge, Qbej, follows equations standard for SPICE, modified to allow specification of a minimum capacitance CEMIN (corresponding to reach-through to an n+ layer). It should be noted that (as studied by Chris Grossman) there is frequently an extra component of charge storage at the base-emitter heterojunction of HBTs, associated with a minimum in the conduction band energy profile. We have chosen to include this contribution in the transit base delay, as discussed below.

Qbej is computed with the following algorithm:

Define Vmin= VJE*[1-(CJE/CEMIN)^(1/MJE)] (the critical voltage for attaining the minimum capacitance value)

If Vbei<FCE*VJE and Vbei<Vmin:

Qbej=CEMIN*(Vbei-VJE)+CEMIN*VJE*MJE/(MJE-1) *(CJE/CEMIN)^(1/MJE)

Cbej=dQbej/dVbei=CEMIN

If Vbei<FCE*VJE and Vbei>Vmin:

Qbej= -CJE*VJE*(1-Vbei/VJE)^(1-MJE) / (1-MJE)

Cbej= CJE*(1-Vbei/VJE)^(-MJE)

If Vbei>FCE*VJE, and CJE>CEMIN*(1-FCE)^MJE:

Qbej=-CJE*VJE/(1-FCE)^MJE*[(1-FCE)/(1-MJE)+FCE-Vbei/VJE -MJE*(FCE-Vbei/VJE)^2/2/(1-FCE)]

Cbej=CJE/(1-FCE)^MJE*[1+MJE*(Vbei/VJE-FCE)/(1-FCE)]

If Vbei>FCE*VJE, and CJE<CEMIN*(1-FCE)^MJE,

Qbej=CEMIN*(Vbei-VJE)+CEMIN*VJE*MJE/(MJE-1)

*(CJE/CEMIN)^(1/MJE)+CJE*VJE*(Vbei/VJE-FCE)^2*MJE/2/

(1-FCE)^(MJE+1)

Cbej=CEMIN + CJE*VJE*MJE*(Vbei/VJE-FC)/(1-FCE)^(MJE+1)

Parameters: CJE, MJE, VJE, CEMIN, FCE. As discussed below, CJE and VJE are allowed to vary with temperature (Top); the values employed in the above formulas are the temperature-corrected values.

Base-Emitter Diffusion Charge, Qbediff:

Diffusion charge in HBTs is associated with contributions from minority carriers in the base and from mobile charge in the collector depletion region. In homojunction transistors, diffusion charge storage in the emitter is also present. In the present model, the base and collector-depletion region contributions are considered separately (if necessary, the emitter charge storage can be associated with the base contribution).

The base charge is specified through the base transit time TFB. This transit time varies with bias through several mechanisms: 1) the Early effect causes a change in transit time with junction voltage; 2) in heterojunction transistors, there is frequently a minimum in the conduction band on the base side of the base-emitter (and potentially base-collector) heterojunction. Minority carriers tend to accumulate in these potential wells. The stored charge adds to the base charge (to a good approximation). To lowest order, the charge stored is directly proportional to the collector current, and thus is accountable with a contribution to TFB. To a greater degree of accuracy, the depth of the potential well on the emitter side varies with Vbe. Similarly, the amount of charge stored at the base-collector side varies with Vbc. The equations used to describe the effects are:

TFBt=TFB*(1+Vbei/VAR+Vbci/VAF) + TBEXS*exp(-q(Vbei-VJE)/NA/KTop) + TBCXS*exp(q(Vbci-VJC)/NB/KTop)

Note that the signs associated with BE and BC junction effects are different. The temperature T to describe these effects is taken to be Top (the estimated junction temperature, rather than the self-consistent one).

The collector charge is specified through three separate mechanisms:

1) a part is specified by the transit time parameter TFC0, modified by the velocity modulation factor qcc to account for voltage and current dependences;

2) a part of the mobile charge is specified in the calculation of base-collector depletion region charge. To calculate this part, Qbcm, an expression for the collector current-dependent base-collector depletion charge is developed; then the current-independent part is subtracted off (as discussed in the section below). Finally,

3) a separate charge term, Qkrk, associated with the Kirk effect is included.

Qfdiff = Icf*ftt*(TFBt + TFC0/qcc) + Qbcm + Qkrk

Here ftt is a factor accounting for temperature dependence of transit times (calculated with the self-consistent device temperature Td):

ftt=rTd^XTTF

and qcc is a factor describing bias dependence of electron velocity in the BC depletion region:

qcc= [1 + (Icf/ITC)^2] / [1 + (Icf/ITC2)^3 + (VJCI-Vbci)/VTC]

ITC is the threshold current for the velocity profile modulation effect, and ITC2 is a higher current at which the velocity profile modulation peaks (and the cutoff frequency begins to roll-off). VTC provides a voltage (or electric field) dependence of the carrier velocity.

The temperature dependence of the parameters is calculated self-consistently as

ITC=ITC@Tnom* rTd^XTITC

ITC2=ITC2@Tnom* rTd^XTITC2

The charge storage associated with the Kirk effect is calculated with the expression:

Qkrk=TKRK*Icf*exp[Vbci/VKRK+Icf/IKRK]

To account for excess phase, a fraction (1-FEX) of the current-dependent forward charge Qfdiff is associated with the BE junction, while the remainder is associated with the intrinsic BC junction.

Qbediff= (1-FEX)*Qfdiff

It should be noted that Qfdiff (and thus Qbediff) depends on Vbci through the terms involving Icf, qcc, Qkrk and Qbcm. As a result, a transcapacitance is implied in the ac model. Similarly, Qbcdiff depends on Vbei, implying another transcapacitance.

The parameters needed to specify Qfdiff include FEX, TFB, TBEXS, TBCXS, TFC0, ITC, ITC2, VTC, TKRK, VKRK and IKRK. The temperature dependences of TFB, TFC0, ITC and ITC2 are determined with the self-consistent temperature through XTTF, XTITC and XTITC2 as noted above, while TKRK, VKRK and IKRK are calculated with the operating temperature.

Intrinsic Base-Collector Charge, Qbci:

Charge stored at the intrinsic base-collector junction includes depletion charge from the junction region, as well as diffusion charge associated with normal operation of the transistor, and diffusion charge associated with reverse operation of the device.

Qbci=Qbcj + TRI* Icr + FEX*Qfdiff

Intrinsic base-collector depletion charge Qbcj:

While the charge in the depletion region is dependent on Ic, we consider in this section the portion corresponding to the condition Ic=0. Subsequently, the proper Ic dependent contribution will be considered, and included in Qbcm (a charge that is part of Qfdiff ) .

With Ic=0, the depletion charge is calculated with the same algorithm as applied to Qbej (which accounts for a minimum of capacitance when the n- collector is depleted):

Define Vmin= VJC*[1-(CJC/CCMIN)^(1/MJC)]

If Vbci<FC*VJC and Vbci<Vmin:

Qbcj=CCMIN*(Vbci-VJC)+CCMIN*VJC*MJC/(MJC-1) *(CJC/CCMIN)^(1/MJC)

Cbcj=CCMIN

If Vbci< FC*VJCI and Vbci>Vmin:

Qbcj= -CJC*VJC*(1-Vbci/VJC)^(1-MJC) / (1-MJC)

Cbcj= CJC*(1-Vbci/VJC)^(-MJC)

If Vbci>FC*VJC, and CJC>CCMIN*(1-FC)^MJC:

Qbcj=-CJC*VJC/(1-FC)^MJC*[(1-FC)/(1-MJC)+FC-Vbci/VJC-MJC*(FC-Vbci/VJC)^2/2/(1-FC)]

Cbcj=CJC/(1-FC)^MJC*[1+MJC*(Vbci/VJC-FC)/(1-FC)]

If Vbci>FC*VJC, and CJC<CCMIN*(1-FC)^MJC:

Qbcj=CCMIN*(Vbci-VJC)+CCMIN*VJC*MJC/(MJC-1)*(CJC/CCMIN)^(1/MJC)+CJC*VJC*(Vbci/VJC-FC)^2*MJC/2/(1-FC)^(MJC+1)

Cbcj=CCMIN + CJC*VJC*MJC*(Vbci/VJC-FC)/(1-FC)^(MJC+1)

Parameters: CJC, MJC, VJC, CCMIN, FC. As discussed below, CJC and VJC are allowed to vary with temperature (Top); the values associated with the above formulas are the temperature-corrected values.

Intrinsic base-collector diffusion charge:

For reverse operation, a diffusion capacitance is implied by the TRI term in the Qbci equation. Here TRI is the effective reverse transit time, taken to be bias independent. The associated reverse diffusion capacitance is

Cbcrdiff= TRI* dIbci/dVbci

For forward operation, diffusion capacitance is also included, in a manner similar to base-emitter capacitance, with a partioning specified by the excess phase factor FEX. The terms associated with Icf*ftt*(TFB + TFC0/qcc) + Qkrk have already been discussed above for the calculation of Qbediff. We now consider the portion Qbcm.

Qbcm:

This charge is defined to be the difference between the "proper" Icf dependent charge in the BCi depletion region, which we call Qbcf, and the BCi depletion charge computed above (Qbcj) under the assumption Icf=0.

Qbcm=Qbcf - Qbcj

In order to properly compute Qbcf, a formulation of depletion region charge similar to that used above is used, with the modification that the CJ parameter (zero bias capacitance) is allowed to be dependent on collector current Icf. This corresponds to the physical phenomenon of varying charge density in the depletion region, as a result of the mobile electron charge in that region.

The current-dependent CJ parameter is termed CJCH, and has the form

CJCH= CJC*sign(1-Icf/ICRIT)*ABS(1-Icf/ICRIT)^MJC

Here ICRIT is a critical current, at which the effective charge density in the BC depletion region vanishes (and the capacitance Cbci drops dramatically). ICRIT is dependent on temperature and bias conditions according to

ICRIT=ICRIT0*qcc/ftt

where ftt and qcc are the temperature-dependence, and Icf and Vcb dependence parameters given above.

Given CJCH, the charge in the depletion region is given by an expression similar to what is used for current-independent cases, generalized to accomodate the possibility of negative CJCH. The algorithm for the charge Qbcf is :

If CJCH<0,

Qbcj=CCMIN*(Vbci-VJC)- CCMIN*VJC*MJC/(MJC-1)*(- CJCH/CCMIN)^(1/MJC)

Cbcj=CCMIN

If CJCH>0,

Define Vmin= VJC*[1-(CJCH/CCMIN)^(1/MJC)]

If Vbci<FC*VJC and Vbci<Vmin:

Qbcj=CCMIN*(Vbci-VJC)+CCMIN*VJC*MJC/(MJC-1) *(CJCH/CCMIN)^(1/MJC)

Cbcj=CCMIN

If Vbci<FC*VJC and Vbci>Vmin:

Qbcj= -CJCH*VJC*(1-Vbci/VJC)^(1-MJC) / (1-MJC)

Cbcj= CJCH*(1-Vbci/VJC)^(-MJC)

If Vbci>FC*VJCI, and CJCH>CCMIN*(1-FC)^MJC,

Qbcj=-CJCH*VJC/(1-FC)^MJC*[(1-FC)/(1-MJC)+FC-Vbci/VJC- MJC*(FC-Vbci/VJC)^2/2/(1-FC)]

Cbcj=CJCH/(1-FC)^MJC*[1+MJC*(Vbci/VJC-FC)/(1-FC)]

If Vbci>FC*VJC, and CJCH<CCMIN*(1-FC)^MJC,

Qbcj=CCMIN*(Vbci-VJC)+CCMIN*VJC*MJC/(MJC-1)*(CJCH/CCMIN)^(1/MJC) +CJCH*VJC*(Vbci/VJC-FC)^2*MJC/2/(1-FC)^(MJC+1)

Cbcj=CCMIN + CJCH*(Vbci/VJC-FC)*MJC/(1-FC)^(MJC+1)

Expressions for charge and capacitance differ in various regions of the CJCH, Vbc plane. Fig. 3 illustrates the regions in which the different expressions are applicable.

With this formulation, the current dependence of the BC capacitance is included (although it is partially assigned to the BE junction charge, and partially to the BC junction, through the parameter FEX, excess phase). ICRIT and associated parameters may be extracted from measurements of Cbc vs Ic. It must be noted that the parameters also control some of the components of the forward transit time. There is a delay time associated with the specification of ICRIT, given by

TFC1 = CJC*VJC*MJC/(MJC-1)/ICRIT

Thus the selection of the parameter ICRIT must be done with care, generally in conjunction with the selection of TFC0 and CJCI, in such a way that the sum TFB + TFC0 + TFC1 provide a reasonable estimate of what in Gummel-Poon SPICE is lumped into TF.

Excess phase

The objective is to evaluate the collector current as Icf (t-td) (a delayed version of Icf(t), where td is a delay time not accounted for with the simple Cbe/gm pole). Conventional Spice uses the Weil formalism of Bessel polynomial computed with backward euler integration. Formulation based on transcapacitances has been proposed. In this model, the charge associated with the base and base-collector depletion regions can be partitioned between the BE junction and intrinsic BC junction, according to the specification of FEX (which defaults to 1).

Extrinsic Base-Collector Charge, Qbcx:

The stored charge Qbcx consists of a depletion charge and a diffusion charge. The diffusion charge component is not considered in standard SPICE, but can constitute an important contribution to saturation stored charge in many HBTs (in addition to the contribution associated with the intrinsic base-collector junction). The corresponding charge storage time TRX may be different from the intrinsic time TRI, because of implant-induced recombination, surfaces, or other structural changes. The depletion charge is taken to correspond to a standard depletion region expression (without considering charge denisty modulation due to current), modified to allow for a minimum value of capacitance under the condition of reach-through. Furthermore, as indicated below, if the variable XCJC is assigned a value different from unity, then the depletion charge is partitioned between the Bx-Cx capacitance and the B-Cx capacitance.

Qbcx= TRX*Ibcx + XCJC*Qbcxo

Here Qbcxo is the depletion charge, computed with the following algorithm:

Define Vmin= VJCX*[1-(CJCX/CXMIN)^(1/MJCX)]

If Vbcx<FC*VJCX and Vbcx<Vmin:

Qbcxo=CXMIN*(Vbcx-VJCX)+CXMIN*VJCX*MJCX/(MJCX-1)*(CJCX/CXMIN)^(1/MJCX)

Cbcxo=CXMIN

If Vbcx<FC*VJCX and Vbcx>Vmin:

Qbcxo= -CJCX*VJCX*(1-Vbcx/VJCX)^(1-MJCX) / (1-MJCX)

Cbcxo= CJCX*(1-Vbcx/VJCX)^(-MJCX)

If Vbcx>FC*VJCX, and CJCX>CXMIN*(1-FC)^MJCX:

Qbcxo=-CJCX*VJCX/(1-FC)^MJCX*[(1-FC)/(1-MJCX)+FC-Vbcx/VJCX-MJCX*(FC-Vbcx/VJCX)^2/2/(1-FC)]

Cbcxo=CJCX/(1-FC)^MJCX*[1+MJCX*(Vbcx/VJCX-FC)/(1-FC)]

If Vbcx>FC*VJCX, and CJCX<CXMIN*(1-FC)^MJCX:

Qbcxo=CXMIN*(Vbcx-VJCX)+CXMIN*VJCX*MJCX/(MJCX-1)*(CJCX/CXMIN)^(1/MJCX) + CJCX*VJCX*(Vbcx/VJCX-FC)^2*MJCX/2/(1-FC)^(MJCX+1)

Cbcxo=CXMIN + CJCX*VJCX*MJCX*(Vbci/VJCX-FC)/(1-FC)^(MJCX+1)

As a result the dependences of Ibcx on Vbcx, a diffusion capacitance results from the formulation:

Cbcxdiff= TRX* dIbcx/dVbcx

Parameters required: TRX, CJCX, MJCX, VJCX, CXMIN

Base-Extrinsic Collector Charge Qbcxx, and Treatment of XCJC:

In standard Spice, XCJC indicates the fraction of overall Cbc depletion capacitance that should be associated with the intrinsic base node, with the remaining fraction (1-XCJC), attached to the base terminal. In HBT Spice, we use a similar assignment: the depletion charge associated with the extrinsic base-collector junction is partiitioned between the Bx node and the B node:

Qbcx= TRX*Ibcx + XCJC* Qbcxo

has been defined above, between the Bx and Cx nodes, and charge

Qbcxx= (1-XCJC)*Qbcxxo

is assigned between the B and Cx nodes. The charge Qbcxxo is computed with the same algorithm as for Qbcxo, using the voltage Vbcxx rather than Vbcx.

According to our formulation, if XCJC is different from unity, there will be 3 contributions to Qbc: Qbci, Qbcx and Qbcxx, and 3 corresponding capacitances.

Parameter required: XCJC.

Collector-Substrate Charge, Qcs:

This corresponds to a depletion charge, formulated in the standard SPICE fashion:

For Vcs>-FC*VJS,

Qcs= - CJS*VJS*(1+Vcs/VJS)^(1-MJS) / (1-MJS)

Ccs=CJS*(1+Vcs/VJS)^(-MJS)

For Vcs<-FC*VJS,

Qcs= -CJS*VJS/(1-FC)^MJS* [(1-FC)/(1-MJS) + FC +Vcs/VJS -MJS/2/(1-FC) *(FC+Vcs/VJS)^2]

Ccs=CJS*(1-FC)^(-MJS)*[1-MJS/(1-FC)*(FC+Vcs/VJS)}

Parameters required: CJS, MJS, VJS.

Thermal Circuit

A thermal subcircuit is used to compute a self-consistent (or "dynamic") temperature Td for the device if the flag SELFT is set to be "true". For this case, Td is equal to the variable T in the equations below. Otherwise, for SELFT="false", the temperature is taken to be the instance parameter Top for all evaluations. Top can be specified by the user for each transistor; if not explicitly stated, Top defaults to global temperature (which defaults to 27C).

The thermal subcircuit is based on the equivalence I<-> Power and V<-> Temperature. The current (power) source is the instantaneous electrical power dissipation in the transistor associated with the non-energy storage elements. The source connects the node Tcommon (a global thermal ground, corresponding to absolute zero temperature) to the node T (which corresponds to the transistor internal temperature). While T is not an external node, the Spice program should store the parameter and allow it to be printed and plotted.

The transistor has a single thermal resistance and thermal capacitance connecting it to an external temperature node, Th. The Th node typically corresponds physically to a heatsink, but can be used to construct thermal circuits where heating of one transistor by another takes place, or to construct elaborate self-heating models (where, for example, a better approximation than a single pole can be used to describe the time dependence of device temperature). If the user does not specify a network to determine Th, then Th will be set equal to the global temperature specified (which in turn defualts to 27C).

Ith= Icc*(Vbei-Vbci) + Ibei*Vbei + Ibex*Vbex + Ibcx*Vbcx + (Ibci-Ibk)*Vbci

+ (V(B)-V(Bx))^2/RBX + (V(Bx)-V(Bi))^2/RBI+ (V(Ei)-V(E))^2/RE

+ (V(C)-V(Cx))^2/RCX + (V(Cx)-V(Ci))^2/RCI - Ics*Vcs

CTH * d T/dt = Ith - (T-Th)/RTH

CTH is independent of voltages, currents, and instantaneous temperature. RTH varies with temperature according to a power law as described below. RTH is computed with the self-consistent or dynamic temperature Td.

Temperature Scaling

In the computation of thermal effects, there is a choice to be made about whether to update continuously the temperature dependence of a parameter while determining self-heating effects (which is computationally intensive but accurate), or to evaluate the parameter only at a single representative temperature (judiciously chosen) for the entire circuit computation. In this model, both of these procedures are used (for different variables). Thus we distinguish between the device temperature Td, the computed device temperature utilizing self-heating, determined self-consistently, and the "operating temperature" Top, which can be specified in the device instance card, which corresponds to the best estimate of the designer about the eventual temperature of the device within the circuit - not computed self-consistently.

Temperature effects are presented with the following notation:

Tnom: nominal absolute temperature for the specification of HBT parameters

Tcom: absolute temperature common (ground) for the thermal circuits of the overall network

Th: heatsink absolute temperature for the device.

Td: device absolute temperature, self-consistently determined. The following abbreviations are also used:

Vtd=K*Td/q

rTd=Td/Tnom

The device node T included in the model corresponds to Td. (This leads to some notation ambiguity).

Top: operating absolute temperature (not modified within the program)

Vtop=K*Top/q

rTop=Top/Tnom

Parameters that are computed using Td include IS, ISA, ISB, ISE, ISEX, ISC, NE, NC, BF, BR, TFB, TFC, ITC, ITC2 and RTH.

Parameters that are computed using Top include most of the remaining temperature dependent quantitites listed below.

* Current flow contributions

IS=IS@Tnom* rTd^(XTI)*exp[EG/(NF*Vtd)*(rTd-1) ]

ISA=ISA@Tnom* rTd^(XTI)*exp[(EG/(NF*Vtd)+EAA/Vtd)*(rTd-1)] ISB=ISB@Tnom* rTd^(XTI)*exp[(EG/(NF*VTd)+EAB/Vtd)*(rTd-1)]

BF(T)=BF@Tnom*rTd^XTB

BR(T)=BR@Tnom*rTd^XTB

(Note:the parameter XTB is allowed to be positive or negative.)

ISE=ISE@Tnom*rTd^(-XTB+XTI/NE)*exp[(EG/NE/VTd- EG/NEo/KTnom+EAE/VTd)*(rTd-1)]

NE=NE@Tnom+TNE*(Td-Tnom)=NEo+TNE*(Td-Tnom)

ISEX=ISEX@Tnom*rTd^(-XTB+XTI/NEX)*exp[(EG/NEX/VTd- EG/NEXo/KTnom+EAX/VTd)*(rTd-1)]

NEX=NEX@Tnom+TNEX*(T-Tnom)=NEXo+TNEX*(Td-Tnom)

ISC=ISC@Tnom*rTd^(-XTB+XTI/NC)*exp[(EG/NC/VTd - EG/NCo/+EAC/VTd)*(rTd-1)]

NC=NC@Tnom+TNC*(Td-Tnom)=NCo+TNC*(Td-Tnom)

The temperature dependence of Ic may be accounted for through XTI, EG, EAA and EAB alone.

The temperature dependence of Ib may be accounted for by EAE alone. For the convenience of users, XTB is also provided (which generally would not be used in conjunction with EAE). Similarly, the ideality factors NE and NEX are allowed to be temperature dependent (but in general a self-consistent formalism could be generated without requiring this variation).

* Parasitic Resistance

RE=RE@Tnom * rTop^XRE

REX=REX@Tnom*rTop^XREX

RBX=RBX@Tnom * rTop^XRB

RBI=RBI@Tnom * rTop^XRB

RCX=RCX@Tnom*rTop^XRC

RCI=RCI@Tnom*rTop^XRC

* Capacitances

VJi=VJi@Tnom - TVJi*(Top-Tnom)

CJi=CJi@Tnom * (VJi@Tnom/VJi)^MJi for i=E,CI,CX,S

MJi, FC,CiMIN,FC, XCJC are temperature independent

*Time delays

TFB and TFC depend on temperature through ftt, which varies according to

ftt=ftt@Tnom*rTd^XTTF

TFC is also dependent on qcc, which varies with temperature through ITC=ITC@Tnom*rTd^XTITC

ITC2=ITC2@Tnom*rTd^XTITC2

TKRK varies with temperature according to

TKRK=TKRK@Tnom*rTop^XTTKRK

and VKRK=VKRK@Tnom*rTop^XTVKRK

IKRK=IKRK@Tnom*rTop^XTIKRK

VTC, FEX are independent of temperature

* Thermal parameters

CTH is temperature independent

RTH =RTH@TNOM*rTd^XRT

* Noise parameters

KFN,AFN,BFN are taken to be temperature independent. The Johnson noise contributions are computed with the self-consistent temperature (although it should be noted that noise analysis is carried out on an "ac" basis, in which temperature is not adjusted self-consistently; the self-consistent temperature Td for noise computations is that which results from the preceding operating point analysis).

The computation of the self-consistent temperature Td is frequently a numerically intensive task that can lead to long computation times or convergence difficulties. Convergence may be improved by artificially limiting (or "clipping") excursions of trial temperature during computation. For negative excursions, Tdev can be clipped at Th (cooling "never" occurs). For positicve excursions, the user can specify the model parameter DTMAX, indicating that T can be restricted to never go above Th+DTMAX.

Noise

Noise current generators are included in the model in fashion similar to standard Spice. The noise current generators have magnitudes in units of A^2/Hz, and computations are done on the basis of 1Hz bandwidth. The noise sources are placed in parallel with corresponding linearized elements in the small signal model. Sources of 1/f noise have magnitudes that vary with frequency f; an exponent BFN is allowed in case f^-1 behavior is not exactly observed.

inc2 = 2*q*Icc

inb2= 2 * q *Ibe + KFN * Ibe^AFN / f^BFN

inre2= 4 * K* Td / RE

inrbx2= 4 * K * Td / RBX

inrbi2= 4 * K * Td / RBI

inrcx2= 4 * K * Td / RCX

inrci2= 4 * K * Td / RCI

inrex2= 4 * K * Td / REX

Substrate Interactions

In this model, the only interactions with the substrate are provided by the substrate-extrinsic collector diode and the associated capacitance Ccs.

we are thus ignoring parasitic pnp specification.

Size Scaling

Scaling of parameters according to specification of device area is supported, as in conventional BJT model. There is no scaling associated with device perimeter. Scaling is introduced in conventional fashion: for area factor A,

*currents, charges and capacitances scale as A

*resistances scale as 1/A

*N,VJ, MJ, Beta, TF components,VA, activation energies and exponents for T variation remain unchanged

6.0 Figures

Fig. 1. Circuit diagram for large signal HBT model

Fig. 2. Circuit diagram for small signal HBT model

Fig. 3. Base-collector voltage - collector current plane, showing different regions for the calculation of base-collector charge and capacitance

8.0 Appendix B

This appendix reviews the equations of the HBT model, discussing them in an order convenient for implementation of SPICE code. The branch currents within the model are discussed in sequence, with proper account of their derivatives. The present version corrects some bugs that have been noted in the equations listed previously. The equations represented here correspond with code that has successfully executed.

Nodes:

In the notation of this model presentation, the nodes are E,Ei,Ex,B,Bi,Bx,C,Ci,Cx,S,T,and Th (for the case SELFT="true").

For the case SELFT="false", the nodes T and Th are not defined (and Top is used in place of T in the equations below).

Vbei=VBi-VEi

Vbci=VBi-VCi

Vbcx=VBx-VCx

Vbcxx=VB-VCx

Vcx=VCx-VS

Current Flow Elements

* Icc

Current from Ci node to Ei node.

Dependences: Vbei, Vbci, T

* Ibei

Current from Bi node to Ei node.

Dependences: Vbei, Vbci, T

* Ibci

Current from Bi node to Ci node

Dependences: Vbei, Vbci, T

* Ibk

Current from Ci node to Bi node

Dependences: Vbei, Vbci, T

* Ibex

Current from Ex node to Ei node

Dependences: VEx-VEi, T

* Ics

Current from S node to Cx node

Dependences: Vcs

* Ibcx

Current from Bx node to Cx node

Dependences: VBx-VCx

*Ith

Current from Tcomm node to T node

Dependences: Vbei, Vbci, VEi-VE, VBx-VBi, VB-VBx, VCx-VCi, VC-VCx,VBx-VEx, VEx-VEi, VCx-VS, T

*Rth

1/Conductance from Node T to Node Th

Dependences: T

* Rcx

1/Conductance from node C to node Cx

Dependences: none

* Rci

1/Conductance from node Ci to Cx

Dependences: none

* Rbx

1/conductance from node B to node Bx

Dependences: none

* Rbi

1/conductance from node Bx to node Bi

Dependences: none

* Re

1/conductance from node Ei to node E

Dependences: none

* Rex

1/conductance from node Ex to Ei

Dependences: none

******** T scaling of parameters

* To assist convergence, T is checked to determine that it is not unrealistically low (below Th) or unrealistically high (above Th+DTMAX). If either limit is exceeded, T is set to the limiting value.

* The instanceT is set equal to T, in order to allow readout of T as a variable after the computation is complete.

* Abbreviations: delt=T-Th

vTd= q/K/T

rTd=T/Tnom

IS=IS@Tnom*exp[q*EG/KT*(rTd-1)]*rTd^XTI

ISA=ISA@Tnom*exp[q*(EG+EAA)/KT*(rTd-1)]*rTd^XTI

ISB=ISB@Tnom*exp[q*(EG+EAB)/KT*(rTd-1)]*rTd^XTI

1/IS*dIS/dT=XTI/T + q*EG/KT /T

1/ISA*dISA/dT=XTI/T + q*(EG+EAA)/KT) / T

1/ISB*dISB/dT=XTI/T + q*(EG+EAB)/KT) / T

BF=BF@Tnom*rTD^XTB

ISE=ISE@Tnom*rTd^(-XTB+XTI/NE)*exp[q*(EG+EAE)/NE/KT- q*(EG+EAE)/NEo/KTnom]

NE=NE@Tnom+TNE*(T-Tnom)=NEo+TNE*(T-Tnom)

(1/BF)*dBF/dT = XTB/T

(1/ISE)*dISE/dT=(XTI/NE-XTB - (EG+EAE)/NE/KT+(EG+EAE)*TNE*T/NE^2/KT-XTI*TNE*T/NE^2*ln(rTd))/ T

BR=BR@Tnom*rTd^XTB

ISC=ISC@Tnom*rTd^(-XTB+XTI/NC)*exp[q*(EG+EAC)/NC/KT- q*(EG+EAC)/NCo/KTnom]

NC=NC@Tnom+TNC*(T-Tnom)=NCo+TNC*(T-Tnom)

(1/BR)*dBR/dT = XTB/T

(1/ISC)*dISC/dT=(XTI/NC-XTB -q*(EG+EAC)/NC/KT+q*(EG+EAC) *TNC*T/NC^2/KT - XTI*TNC*T/NC^2*ln(rTd)) / T

ISEX=ISEX@Tnom*rTd^(-XTB+XTI/NEX)*exp[q*(EG+EAX)/NEX/KT- q*(EG+EAX)/NEXo/KTnom]

NEX=NEX@Tnom+TNEX*(T-Tnom)=NEXo+TNEX*(T-Tnom)

(1/ISEX)*dISEX/dT=(XTI/NEX-XTB -q*(EG+EAX) /NEX/KT+ q*(EG+EAX)*TNEX*T/NEX^2/KT-XTI*TNEX*T/NEX^2*ln(rTd) )/ T

****** Icc

Current from Ci node to Ei node.

Dependences: Vbei, Vbci, T

Icc=Icf-Icr

Icf

Icf= IS * [ exp(qVbei/NF/ KT) - 1 ] / D

D= qb + IS*exp(qVbei/NA/KT)/ISA + IS* exp(qVbci/NB/KT)/ISB

qb= q1*[1+(1+4*q2)^0.5]/2

q1 = [ 1 - Vbci/VAF - Vbei/VAR ]^-1

q2= IS/IK*[exp(qVbei/NF/KT)-1]

dIcf/dVbei = + (Icf + IS/D)*q/NF/KT - (Icf/D) dD/dVbei

where dD/dVbei = +qb*q1/VAR

+ q1*IS/IK*(q/NF/KT) *exp(qVbei/NF/KT) /(1+4q2)^0.5 + (IS/ISA)*exp(qVbei/NA/KT)*(q/NA/KT)

dIcf/dVbci=- (Icf/D)*dD/dVbci

where dD/dVbci= qb*q1/VAF + (IS/ISB)*exp(qVbci/NB/KT)*(q/NB/KT)

dIcf/dT = Icf*(1/IS*dIS/dT)-(Icf/D)*dD/dT- (IS/D)*exp(qVbei/NF/KT)*(qVbei/NF/KT)/T

dD/dT=

q1/(1+4*q2)^0.5*IS/IK*exp(qVbei/NF/KT)*[-qVbei/NF/KT/T +(1/IS)* dIS/dT] +

(IS/ISA)*exp(qVbei/NA/KT)*[(1/IS*dIS/dT )- (1/ISA*dISA/dT) -(qVbei/NA/KT)/T]

+(IS/ISB)*exp(qVbci/NB/KT)*[(1/IS*dIS/dT) - (1/ISB*dISB/dT) - (qVbci/NB/KT)/T]

Icr

Icr= IS * [ exp(qVbci/NR/KT) - 1 ] / D

and (as given above)

D= qb + IS*exp(qVbei/NA/KT)/ISA + IS* exp(qVbci/NB/KT)/ISB

dIcr/dVbei = - (Icr/D) dD/dVbei

where dD/dVbei is given above

dIcr/dVbci= (Icr + IS/D)*q/(NR/KT) - (Icr/D)*dD/dVbci

where dD/dVbci is given above

dIcr/dT = (Icr/IS)*dIS/dT - (Icr/D)*dD/dT - (IS/D)*exp(qVbci/NR/KT)*(qVbci/NR/KT) / T

where 1/IS*dIS/dT is given above and dD/dT is given above.

***** Ibei

Current from B node to Ei node.

Dependences: Vbei, Vbci, T

Ibei= Icf/BF + ISE * [exp(qVbei/NE/KT) - 1]

dIbei/dVbei = (1/BF)*dIcf/dVbei + ISE*exp(qVbei/NE/KT)*(q/NE/KT)

with dIcf/dVbei given above

dIbei/dVbci= (1/BF)*dIcf/dVbci

with dIcf/dVbci given above

dIbei/dT = (Icf/BF)*[(1/Icf)*dIcf/dT - (1/BF)*dBF/dT] + dISE/dT*[exp(qVbei/NE/KT)-1]

- ISE*exp(qVbei/NE/KT)*qVbei*(NE*K+TNE*KT)/(NE*KT)^2

****** Ibci

Current from Bi node to Ci node

Dependences: Vbei, Vbci, T

Ibci= Icr/BR + ISC * [exp(qVbci/NC/KT) - 1]

dIbci/dVbei = (1/BR)*dIcr/dVbei

with dIcr/dVbei given above

dIbci/dVbci= (1/BR)*dIcr/dVbci + ISC*exp(qVbci/NC/KT)*(q/NCKT)

with dIcr/dVbci given above

dIbci/dT=(Icr/BR)*[(1/Icr)*dIcr/dT - (1/BR)*dBR/dT] + dISC/dT*[exp(qVbci/NC/KT)-1]

- ISC*exp(qVbci/NC/KT)*qVbci*(NC*K+TNE*KT)/(NC*KT)^2

****** Ibk

Current from Ci node to Bi node

Dependences: Vbei, Vbci, T

If BKDN= false, Ibk=0 (and all derivatives of Ibk=0)

If BKDN=true, Ibk= (Mf-1)*Icf

If BKDN=true,

for Vt<-Vbci<FA*BVC

Mf=1/(1- (-Vbci/BVC)^NBC)

dMf/dVbci= - NBC*Mf^2*(-Vbci/BVC)^(NBC-1) /BVC

for Vt>-Vbci

Mf=1

dMf/dVbci= 0

for -Vbci>FA*BVC

Mf=Mf1+g1*(-Vbci-FA*BVC)

with g1=MF1*(MF1-1)*NBC/FA/BVC

dMf/dVbci= - MF1*(Mf1-1)*NBC/(FA*BVC)

Here Mf1 = 1/(1-FA^NBC)

dIbk/dVbei = (Mf-1) *dIcf/dVbei

with dIcf/dVbei given above

dIbk/dVbci = (Mf-1)* dIcf/dVbci + Icf*dMf/dVbci

dIbk/dT = (Mf-1) *dIcf/dT

with dIcf/dT given above

******* Ibex

Current from Ex node to Ei node

Dependences: VEx-VEi, T

Ibex= ISE X* [exp(q(VEx-VEi)/NEX/KT) - 1]

dIbex/dVEi = - ISEX*exp(q(VEx-VEi)/NEX/KT)*(q/NEX/KT)

dIbex/dVEx = + ISEX*exp(q(VEx-VEi)/NEX/KT)*(q/NEX/KT)

dIbex/dT = + dISEX/dT*[exp(q(VEx-VEi)/NEX/KT)-1]

- ISEX*exp(q(VEx-VEi)/NEX/KT)*q(VEx-VEi)*(NEX*K+TNEX*KT)/(NE*KT)^2

****** Ics

Current from S node to Cx node

Dependences: Vcs

Ics= ICS* [exp(q*Vcs/NCS/KTop) - 1]

dIcs/dVcs = ICS*exp(q*Vcs/NCS/KTop)*(q/NCS/KTop)

*******Ibcx

Current from Bx node to Cx node

Dependences: Vbcx

Ibcx= ISCX* [exp(q*Vbcx/NCX/KTop) - 1]

dIbcx/dVbcx = ISCX*exp(q*Vbcx/NCX/KTop)*(q/NCX/KTop)

******Ith

Current from Tcomm node to T node

Dependences: Vbei, Vbci, VEi-VE, VBx-VBi, VB-VBx, VCx-VCi, VC-VCx,VBx-VEx, VEx-VEi, T

Ith=Icc*(Vbei-Vbci) + Ibei*Vbei + Ibex*(VBx-VEi) + Ibcx*Vbcx

+ (Ibci-Ibk)*Vbci +(VB-VBx)^2/RBX + (VBx-VBi)^2/RBI + (VEi-VE)^2/RE

+(VC-VCx)^2/RCX + (VCx-VCi)^2/RCI - Ics*Vcs

dIth/dVE = 2*(VE-VEi)/RE

dIth/dVEi= (dIcf/dVEi-dIcr/dVEi)*(Vbei-Vbci) -Icf+Icr+dIbei/dVEi *Vbei- Ibei+dIbex/dVEi*(VBx-VEi)-Ibex+(dIbci/dVEi-dIbk/dVEi)*Vbci+ 2*(VE- VEi)/RE

dIth/dVEx= dIbex/dVEx*(VBx-VEi)

dIth/dVB= 2*(VB-VBx) / RBX

dIth/dVBi= (dIcf/dVBi-dIcr/dVBi)*(Vbei-Vbci) + dIbei/dVBi*Vbei+Ibei + (dIbci/dVBi-dIbk/dVBi)*Vbci + Ibci-Ibk +2(VBx-VBi)/RBI

dIth/dVBx= dIbcx/dVBx*Vbcx + Ibcx + 2(VB-VBx)/RBX + 2(VBx-VBi)/RBI

dIth/dVC= 2*(VC-VCx) / RCX

dIth/dVCi= (dIcf/dVCi-dIcr/VCi)*(Vbei-Vbci) + Icf-Icr +dIbei/dVCi*Vbei +(dIbci/dVCi-dIbk/dVCi)*Vbci - Ibci+Ibk-2*(VCx-VCi)/RCI

dIth/dVCx= dIbcx/dVCx*Vbcx-Ibcx+2*(VC-VCx) / RCX +2*(VCx-VCi)/RCI- dIcs/dVCx*Vcs - Ics

dIth/dVS= dIcs/dVS*Vcs + Ics

dIth/dT= (dIcf/dT-dIcr/dT)*(Vbei-Vbci) + dIbei/dT*Vbei +(dIbci/dT- dIbk/dT)*Vbci

******Rth

1/Conductance from Node T to Node Th

Dependences: T

RTH =RTH@TNOM*rTd^XRT

d(1/RTH)/dT= - XRT/RTH/T

****** Rcx

1/Conductance from node C to node Cx

Dependences: none

RCX=RCX@Tnom*rTop^XRC

****** Rci

1/Conductance from node Ci to Cx

Dependences: none

RCI=RCI@Tnom*rTop^XRC

****** Rbx

1/conductance from node B to node Bx

Dependences: none

RBX=RBX@Tnom * rTop^XRB

****** Rbi

1/conductance from node Bx to node Bi

Dependences: none

RBI=RBI@Tnom * rTop^XRB

****** Re

1/conductance from node Ei to node E

Dependences: none

RE=RE@Tnom * rTop^XRE

****** Rex

1/conductance from node Bx to Ex

Dependences: none

REX=REX@Tnom*rTop^XREX

Charge Storage Elements

* Qbe

Charge beween Bi node (+) and Ei node (-)

Dependences: Vbei, Vbci, T

* Qbci

Charge between Bi node (+) and Cinode (-)

Dependences: Vbei, Vbci, T

*Qbcx

Charge between Bx node and Cx node

Dependences: Vbcx, Vbei

* Qbcxx

Charge between B node (+) and Cx node (-)

Dependences: VB-VCx, Vbei

* Qcs

Charge between Cx node (+) and S node (-)

Dependences: Vcs

* Qth

Charge between T node (+) and Th node (-)

Dependences: T-Th

DepletionCapMod

Numerous charge and capacitance contributions are computed with the subroutine

DepletionCapMod (charge,v,cj,vj,mj,fj,cmin)

defined as follows:

Define vmin= vj*[1-(cj/cmin)^(1/mj)]

If v<fc*vj and v<vmin:

charge=cmin*(v-vj)+cmin*vj*mj/(mj-1)*(cj/cmin)^(1/mj)

cap=cmin

If v<fc*vj and v>vmin:

charge= -cj*vj*(1-v/vj)^(1-mj) / (1-mj)

cap= cj*(1-v/vj)^(-mj)

If v>fc*vj, and cj>cmin*(1-fc)^mj:

charge=-cj*vj/(1-fc)^mj*[(1-fc)/(1-mj)+fc-v/vj-mj*(fc-v/vj)^2/2/(1-fc)]

cap=cj/(1-fc)^mj*[1+mj*(v/vj-fc)/(1-fc)]

If v>fc*vj, and cj<cmin*(1-fc)^mj,

charge=cmin*(v-vj)+cmin*vj*mj/(mj-1)*(cj/cmin)^(1/mj)+cj*vj*(v/vj-fc)^2*mj/2/(1- fc)^(mj+1)

cap=cmin + cj*mj*(v/vj-fc)/(1-fc)^(mj+1)

***** Qbe

Qbe= Qbej + (1-FEX)*Qfdiff

VJi=VJi@Tnom - TVJi*(Top-Tnom)

CJi=CJi@Tnom * (VJi@Tnom/VJi)^MJi for i=E,CI,CX,S

Qbej is a function only of the voltage Vbei, and is computed with :

Qbej=DepletionCapMod(charge=Qbej,v=Vbei,cj=CJE,vj=VJE,mj=MJE,

fc=FCE,cmin=CEMIN)

Qfdiff

Qfdiff is the forward "diffusion" stored charge. Qfdiff may be considered to be a function of the variables Icf ,Vbci and T (which are in turn functions of the node voltages).

Qfdiff= Icf*ftt*(TFBt + TFC0/qcc) + Qbcm + Qkrk

ftt=rTd^XTTF

TFBt=TFB*(1+Vbci/VAF+Vbei/VAR) + TBEXS*exp(-q(Vbei-VJE)/NA/KT)

+ TBCXS*exp(q(Vbci-VJC)/NB/KT))

qcc= [1 + (Icf/ITC)^2] / [1 + (Icf/ITC2)^3 + (VJCI-Vbci)/VTC]

ITC=ITC@Tnom* rTd^XTITC

ITC2=ITC2@Tnom* rTd^XTITC2

Qkrk=TKRK*Icf*exp[Vbci/VKRK+Icf/IKRK]

TKRK=TKRK@Tnom*rTop^XTTKRK

VKRK=VKRK@Tnom*rTop^XTVKRK

IKRK=IKRK@Tnom*rTop^XTIKRK

Qbcm:

Qbcm=Qbcf - Qbcj

Qbcj is a function of Vbci only, as defined below. Qbcf is dependent on collector current as well as voltage. In what follows, it is convenient to compute Qbcf as a function of the variables Vbci and i, where i=Icf/ICRIT. We define Cbcf=dQbcf/dVbci and tbcf=dQbcf/di.

Qbcf and its derivatives are computed from the following algorithm:

CJCH= CJC*sign(1-Icf/ICRIT)*ABS(1-Icf/ICRIT)^MJC

ICRIT=ICRIT0*qcc/ftt

(ftt and qcc correspond to temperature and bias dependences of ICRIT, defined above).

If CJCH<0,

Qbcf=CCMIN*(Vbci-VJC)- CCMIN*VJC*MJC/(MJC-1) *(- CJCH/CCMIN)^(1/MJC)

Cbcf=CCMIN

tbcf=-CJC*VJC*MJC/(MJC-1)*(-CJCH/CCMIN)^((1-MJC)/MJC)

*ABS(1-Icf/ICRIT)^(MJC-1)

If CJCH>0,

Define Vmin= VJC*[1-(CJCH/CCMIN)^(1/MJC)]

If Vbci<FC*VJC, and Vbci<Vmin,

Qbcf=CCMIN*(Vbci-VJC)+CCMIN*VJC*MJC/(MJC-1) *(CJCH/CCMIN)^(1/MJC)

Cbcf=CCMIN

tbcf= - CJC*VJC*MJC/(MJC-1)*(CJCH/CCMIN)^((1-MJC)/MJC)

*ABS(1-Icf/ICRIT)^(MJC-1)

If Vbci<FC*VJC, and Vbci>Vmin,

Qbcf= -CJCH*VJC*(1-Vbci/VJC)^(1-MJC) / (1-MJC)

Cbcf= CJCH*(1-Vbci/VJC)^(-MJC)

tbcf= - CJC*VJC*MJC/(MJC-1)*(1-Vbci/VJC)^(1-MJC)

*ABS(1-Icf/ICRIT)^(MJC-1)

If Vbci>FC*VJC, and CJCH>CCMIN*(1-FC)^MJC,

Qbcf=-CJCH*VJC/(1-FC)^MJC*[(1-FC)/(1-MJC)+FC-Vbci/VJC- MJC*(FC-Vbci/VJC)^2/2/(1-FC)]

Cbcf=CJCH/(1-FC)^MJC*[1+MJC*(Vbci/VJC-FC)/(1-FC)]

tbcf=-Qbcf*mjc /ABS(1-Icf/ICRIT)

If Vbci>FC*VJC, and CJCH<CCMIN*(1-FC)^MJC,

Qbcf=CCMIN*(Vbci-VJC)+CCMIN*VJC*MJC/(MJC-1) *(CJCH/CCMIN)^(1/MJC) + CJCH*VJC*(Vbci/VJC-FC)^2

*MJC/2/(1-FC)^(MJC+1)

Cbcf=CCMIN + CJCH*(Vbci/VJC-FC)*MJC/(1-FC)^(MJC+1)

tbcf=-CJCH*VJC*(Vbci/VJC-FCC)^2*MJC^2/2/(1-FC)^(MJC+1)

/(1-Icf/ICRIT) - CCMIN*VJC*MJC/(MJC-1)

*(CJCH/CCMIN)^(1/MJC)/(1-Icf/ICRIT)

dQfdiff/dIcf=ftt*(TFBt-(TFB/VAR-TBEXS*exp(-q(Vbe-VJE)/NA/KT)*(q/NA/KT))

*Icf/(dIcf/dVEi)+TFC0/qcc-TFC0/qcc^2*Icf*dqcc/dIcf)

+ dQbcm/dIcf + dQkrk/dIcf

dQfdiff/dVbci=- ftt*Icf*TFC0/qcc^2*dqcc/dVbci + dQbcm/dVbci + dQkrk/dVbci

+ftt*TFB*Icf/VAF + ftt*TBCXS*Icf

*exp(q(Vbci-VJC)/NB/KT)*(q/NB/KT)

dQfdiff/dT= (Qfdiff-Qbcm-Qkrk)*XTTF/rTd/Tnom+ dQbcm/dT

Here

dqcc/dIcf=[2*Icf/ITC^2 -3*Icf^2/ITC2^3-Icf^4/ITC^2/ITC2^3 + 2*Icf/ITC^2*(VJCI-Vbci)/VTC]/ [1 + (Icf/ITC2)^3+(VJCI-Vbci)/VTC]^2

dqcc/dVbci= qcc^2*(1+(Icf/ITC)^2)/VTC

dqcc/dT=-2*Icf^2/ITC^2*XTITC/rTd/Tnom/(1+(Icf/ITC2)^3+(VJCI-Vbci)/VTC)

-3*Icf^3/ITC2^3*XTITC2/rTd/Tnom /

(1+(Icf/ITC2)^3+(VJCI-Vbci)/VTC)^2

dQkrk/dIcf= Qkrk*(1/Icf + 1/IKRK)

dQkrk/dVbci= Qkrk/VKRK

dQbcm/dVbci=Cbcf - Cbcj -tbcf*Icf/ICRIT^2*dICRIT/dVbci

where dICRIT/dVbci=ICRIT0/ftt*dqcc/dVbci

dQbcm/dIcf=tbcf*(1/ICRIT-Icf/ICRIT^2*dICRIT/dIcf)

where dICRIT/dIcf= ICRIT0/ftt*dqcc/dIcf

dQbcm/dT= -tbcf*Icf/ICRIT^2*dICRIT/dT

where dICRIT/dT=ICRIT0*qcc/ftt*(-XTTF/rTd/Tnom +dqcc/dT/qcc)

dQbei/dVbei= Cbej - (1-FEX)*dQfdiff/dIcf * dIcf/dVbei

dQbei/dVbci= + (1-FEX)*(dQfdiff/dIcf *dIcf/dVbci + dQfdiff/dVbci)

dQbei/dT= (1-FEX)*(dQfdiff/dT + dQfdiff/dIcf*dIcf/dT)

***** Qbci

Qbci=Qbcj + TRI* Icr + FEX*Qfdiff

Qfdiff is defined above.

VJi=VJi@Tnom - TVJi*(Top-Tnom)

CJi=CJi@Tnom * (VJi@Tnom/VJi)^MJi for i=E,CI,CX,S

Qbcj is a function only of the voltage Vbci, and is computed with :

Qbcj=DepletionCapMod(charge=Qbcj,v=Vbci,cj=CJC,vj=VJC,mj=MJC,

fc=FC,cmin=CCMIN)

Its associated junction capacitance is denoted Cbcj.

dQbci/dVbei= TRI*dIcr/dVbei + FEX*dQfdiff/dIcf*dIcf/dVbei

dQbci/dVbci=Cbcj+ TRI*dIcr/dVbci+FEX*(dQfdiff/dIcf*dIcf/dVbci + dQfdiff/dVbci)

dQbci/dT= TRI*dIcr/dT+FEX*(dQfdiff/dIcf*dIcf/dT+dQfdiff/dT)

****** Qbcx

Qbcx= TRX*Ibcx + XCJC*Qbcxo

VJi=VJi@Tnom - TVJi*(Top-Tnom)

CJi=CJi@Tnom * (VJi@Tnom/VJi)^MJi for i=E,CI,CX,S

Qbcxo is a function only of the voltage Vbcx, and is computed with :

Qbcxo=DepletionCapMod(charge=Qbcxo,v=Vbcx,cj=CJCX,vj=VJCX,mj=MJCX,

fc=FC,cmin=CXMIN)

Cbcx= XJCX*Cbcxo

dQbcx/dVbei= TRX*dIbcx/dVbei

dQbcx/dVbci=TRX*dIbcx/dVbci

dQbcx/dT= TRX*dIbcx/dT

dQbcx/dVbcx=Cbcx

******* Qbcxx

Qbcxx= (1-XCJC)*Qbcxxo

VJi=VJi@Tnom - TVJi*(Top-Tnom)

CJi=CJi@Tnom * (VJi@Tnom/VJi)^MJi for i=E,CI,CX,S

Qbcxxo is a function only of the voltage Vbcxx, and is computed with :

Qbcxxo=DepletionCapMod(charge=Qbcxxo,v=Vbcxx,cj=CJCX,vj=VJCX,mj=MJCX,

fc=FC,cmin=CXMIN)

Cbcxx=(1-XCJC)*Cbcxxo

dQbcxx/dVB=Cbcxx

dQbcxx/dVCx=-Cbcxx

Qcs

VJi=VJi@Tnom - TVJi*(Top-Tnom)

CJi=CJi@Tnom * (VJi@Tnom/VJi)^MJi for i=E,CI,CX,S

Qcs is computed according to conventional LocalDepletionCapacitance algorithm.

For Vcs>-FC*VJS,

Qcs= - CJS*VJS*(1+Vcs/VJS)^(1-MJS) / (1-MJS)

Ccs=CJS*(1+Vcs/VJS)^(-MJS)

For Vcs<-FC*VJS,

Qcs= -CJS*VJS/(1-FC)^MJS * [(1-FC)/(1-MJS) + FC +Vcs/VJS -MJS/2/(1- FC)*(FC+Vcs/VJS)^2]

Ccs=CJS*(1-FC)^(-MJS)*[1-MJS/(1-FC)*(FC+Vcs/VJS)]

Qth

Qth=CTH*(T-Th)

dQth/dT = CTH

dQth/dTh=-CTH

CTH is temperature independent

Summary of Jacobian Matrix Entries

E:

gEE= +1/RE

gEEi= -1/RE

Ei:

gEiE=-1/RE

gEiEi= dIcf/dVbei - dIcr/dVbei + dIbei/dVbei + dIbex/dVbei + 1/RE

gEiEx= -dIbex/dVEx

gEiBi= - dIcf/dVbei + dIcr/dVbei - dIbei/dVbei - dIcf/dVbci + dIcr/dVbci - dIbei/dVbci

gEiCi=+ dIcf/dVbci - dIcr/dVbci

gEiT= - dIcf/dT+ dIcr/dT - dIbei/dT

Ex:

gExEi= - dIbex/dVEi

gExEx= +1/REX + dIbex/dVEx

gExBx= -1/REX

B:

gBB= +1/RBX

gBBx= -1/RBX

Bi:

gBiEi = - dIbei/dVbei - dIbci/dVbei + dIbk/dVbei

gBiBi = dIbei/dVbei + dIbci/dVbei - dIbk/dVbei +dIbei/dVbci + dIbci/dVbci - dIbk/dVbci + 1/RBI

gBiBx = -1/RBI

gBiCi = -dIbei/dVbci - dIbci/dVbci +dIbk/dVbci

gBiT = dIbei/dT + dIbci/dT - dIbk/dT

Bx:

gBxEx= -1/REX

gBxB= -1/RBX

gBxBi= -1/RBI

gBxBx= +1/REX + 1/RBX + 1/RBI + q*Iscx/(NSC*KTop)

gBxCx= - q Iscx/(NSC*KTop)

C:

gCC= +1/RCX

gCCx= -1/RCX

Ci:

gCiEi = - dIcf/dVbei + dIcr/dVbei - dIbk/dVbei +dIbci/dVbei

gCiBi = dIcf/dVbei - dIcr/dVbei + dIbk/dVbei - dIbci/dVbei + dIcf/dVbci - dIcr/dVbci + dIbk/dVbci - dIbci/dVbci

gCiCi = - dIcf/dVbci + dIcr/dVbci - dIbk/dVbci + dIbci/dVbci + 1/RCI

gCiCx = - 1/RCI

gCiT = dIcf/dT- dIcr/dT + dIbk/dT - dIbci/dT

Cx:

gCxBx= - q *Iscx/(NSC*KTop)

gCxC=

gCxCi= -1/RCI

gCxCx= +1/RCI + 1/RCX + q*Iscx/(NSC*KTop) + q*Ics/(NCS*KTop)

gCxS= -q*Ics/(NCS*KTop)

S:

gSCx= -q*Ics/(NCS*KTop)

gSS= +q*Ics/(NCS*KTop)

T:

gTE = -dIth/dVE

gTEi = -dIth/dVEi

gTEx= -dIth/dVEx

gTB=- dIth/dVB

gTBi = -dIth/dVBi

gTBx= -dIth/dVBx

gTC= -dIth/dVC

gTCi =-dIth/dVCi

gTCx=-dIth/dVCx

gTS=-dIth/dVS

gTT = - dIth/dT+1/RTH - XRT*(T-Th)/RTH/T

gTTh = - 1/RTH+ XRT*(T-Th)/RTH/T

Th:

gThT= - 1/RTH +XRT*(T-Th)/RTH/T

gThTh=+1/RTH-XRT*(T-Th)/RTH/T

Notation:

cab: partial derivative of charge stored at node a, with respect to voltage of node b (with remaining node voltages held constant).

Derivatives:

E, Ex, C:

No capacitance contributions

Ei:

cEiEi= + dQbei/dVbei

cEiBi= - dQbei/dVbei - dQbei/dVbci

cEiCi= + dQbei/dVbci

cEiT= - dQbei/dT

B:

cBB= +Cbcxx

cBCx= -Cbcxx

Bi:

cBiEi = - dQbei/dVbei - dQbci/dVbei

cBiBi = dQbei/dVbei + dQbci/dVbei + dQbei/dVbci + dQbci/dVbci

cBiCi = - dQbei/dVbci - dQbci/dVbci

cBiT = dQbei/dT + dQbci/dT

Bx:

cBxEi= - TRX*dIcr/dVbei

cBxBi= TRX*( dIcr/dVbei + dIcr/dVbci)

cBxBx= Cbcx

cBxCi= - TRX*dIcr/dVbci

cBxCx= -Cbcx

cBxT= TRX*dIcr/dT

Ci:

cCiEi = dQbci/dVbei

cCiBi = -dQbci/dVbei - dQbci/dVbci

cCiCi = dQbci/dVbci

cCiT = -dQbci/dT

Cx:

cCxEi= TRX*dIcr/dVbei

cCxBx= -Cbcx

cCxBi= -TRX*( dIcr/dVbei + dIcr/dVbci)

cCxB= -Cbcxx

cCxCi= TRX*dIcr/dVbci

cCxCx= Cbcxx + Cbcx + Ccs

cCxS= -Ccs

cCxT= -TRX*dIcr/dT

S:

cSCx= -Ccs

cSS= +Ccs

T:

cTT = + CTH

cTTh = -CTH

Th:

cThT= -CTH

cThTh= + CTH

Summary of Model Parameters

The following lists the HBT model parameters. New parameters (not specified within Berkeley SPICE) are noted with an asterisk. The total number of parameters is 91 (while Berkeley SPICE has 42). Temperature dependence specifications account for 27 of the parameters of the present HBT model.

Parameter	Significance	Units	Default
*SELFT	flag denoting self-heating should be included	logic	false
*BKDN	flag denoting that BC breakdown should be included	logic	false
*TNOM	temperature at which model parameters are given	C	27
IS	saturation value for forward collector current	A	1e-25
NF	forward collector current ideality factor	-	1
NR	reverse current ideality factor	-	1
*ISA	collector current EB barrier limiting current	A	1e10
*NA	collector current EB barrier ideality factor	-	2
*ISB	collector current BC barrier limiting current	A	1e10
*NB	collector current BC barrier ideality factor	-	2
VAF	forward Early voltage	V	1000
VAR	reverse Early voltage	V	1000
IK	knee current for dc high injection effect	A	1e10
BF	forward ideal current gain	-	10000
BR	reverse ideal current gain	-	10000
ISE	saturation value for nonideal base current	A	1e-30
NE	ideality factor for nonideal forward base current	-	2
*ISEX	saturation value for emitter leakage diode	A	1e-30
*NEX	ideality factor for emitter leakage diode	-	2
ISC	saturation value for intrinsic bc junction current	A	1e-30
NC	ideality factor for intrinsic bc junction current	-	2
*ISCX	saturation value for extrinsic bc junction current	A	1e-30
*NCX	ideality factor for extrinsic bc junction current	-	2
*FA	Factor for specification of avalanche voltage	-	0.9
*BVC	collector-base breakdown voltage BVcbo	V	1000
*NBC	exponent for BC multiplication factor vs voltage	-	8
*ICS	saturation value for collector-substrate current	A	1e-30
*NCS	ideality factor for collector-substrate current	-	2
RE	Emitter resistance	ohm	0
*REX	Extrinsic emitter leakage diode series resistance	ohm	0
RBX	Extrinsic base resistance	ohm	0
RBI	Intrinsic base resistance	ohm	0
*RCX	Extrinsic collector resistance	ohm	0
*RCI	Intrinsic collector resistance	ohm	0
CJE	BE depletion capacitance at zero bias	F	0
VJE	BE diode builtin potential for Cj estimation	V	1.6
MJE	Exponent for voltage variation of BE Cj	-	0.5
CEMIN	Minimum BE capacitance	F	0
FCE	Factor for start of high bias BE Cj approximation	-	0.8
CJC	Intrinsic BC depletion capacitance at zero bias	F	0
VJC	Intrinsic BC diode builtin potential for Cj estimation	V	1.4
MJC	Exponent for voltage variation of Intrinsic BC Cj	-	0.33
*CCMIN	Minimum value of intrinsic BC Cj	F	0
*FC	Factor for start of high bias BC Cj approximation	-	0.8
*CJCX	Extrinsic BC depletion capacitance at zero bias	F	0
*VJCX	Extrinsic BC diode builtin potential for Cj estimation	V	1.4
*MJCX	Exponent for voltage variation of Extrinsic BC Cj	-	0.33
*CXMIN	Minimum extrinsic Cbc	F	0
XCJC	Factor for partitioning extrinsic BC Cj	-	1
*CJS	Collector-substrate depletion capacitance (0 bias)	F	0
*VJS	CS diode builtin potential for Cj estimation	V	1.4
*MJS	Exponent for voltage variation of CS Cj	-	0.5
TFB	Base transit time	S	0
*TBEXS	Excess BE heterojunction transit time	S	0
*TBCXS	Excess BC heterojunction transit time	S	0
*TFC0	Collector forward transit time	S	0
*ICRIT0	Critical current for intrinsic Cj variation	A	1e3
*ITC	Characteristic current for TFC	A	0
*ITC2	Characteristic current for TFC	A	0
*VTC	Characteristic voltage for TFC	V	1e3
*TKRK	Forward transit time for Kirk effect	S	0
*VKRK	Characteristic Voltage for Kirk effect	V	1e3
*IKRK	Characteristic current for Kirk effect	A	1e3
TR	Reverse charge storage time for intrinsic BC diode	S	0
TRX	Reverse charge storage time for extrinsic BC diode	S	0
*FEX	Factor to determine excess phase	-	0
*RTH	Thermal resistance from device to thermal ground	C/W	1e-8
*CTH	Thermal capacitance of device	C/joule0
KFN	BE flicker noise constant	-	0
AFN	BE flicker noise exponent for current	-	1
BFN	BE flicker noise exponent for frequency	-	1
XTI	Exponent for IS temperature dependence	-	2
XTB	Exponent for beta temperature dependence	-	2
*TNE	Coefficient for NE temperature dependence	-	0
*TNC	Coefficient for NC temperature dependence	-	0
*TNEX	Coefficient for NEX temperature dependence	-	0
EG	Activation energy for IS temperature dependence	V	1.5
*EAE	Activation energy for ISA temperature dependence	V	0
*EAC	Activation energy for ISB temperature dependence	V	0
*EAA	Added activation energy for ISE temp dependence	V	0
*EAB	Added activation energy for ISC temp dependence	V	0
*EAX	Added activation energy for ISEX temp dependence	V	0
*XRE	Exponent for RE temperature dependence	-	0
*XREX	Exponent for REX temperature dependence	-	0
*XRB	Exponent for RB temperature dependence	-	0
*XRC	Exponent for RC temperature dependence	-	0
*TVJE	Coefficient for VJE temperature dependence	V/C	0
*TVJCX	Coefficient for VJCX temperature dependence	V/C	0
*TVJC	Coefficient for VJC temperature dependence	V/C	0
*TVJS	Coefficient for VJS temperature dependence	V/C	0
*XTITC	Exponent for ITC temperature dependence	-	0
*XTITC2	Exponent for ITC2 temperature dependence	-	0
XTTF	Exponent for TF temperature dependence	-	0
*XTTKRK	Exponent for TKRK temperature dependence	-	0
*XTVKRK	Exponent for VKRK temperature dependence	-	0
*XTIKRK	Exponent for IKRK temperature dependence	-	0
*XRT	Exponent for RTH temperature dependence	-	0
*DTMAX	Maximum expected temperature rise above heatsink	C	1000

Sample Data Sets and Fitting

The following parameters describe a representative AlGaAs/GaAs HBT at room temperature.

Parameter	Significance	Value
*SELFT	flag denoting self-heating should be included	true
*BKDN	flag denoting that BC breakdown should be included	false
*TNOM	temperature at which model parameters are given	25C
IS	saturation value for forward collector current	8.36e-26 A
NF	forward collector current ideality factor	1
NR	reverse current ideality factor	1
*ISA	collector current EB barrier limiting current	2.18e-18 A
*NA	collector current EB barrier ideality factor	4.51
*ISB	collector current BC barrier limiting current	1e10 A
*NB	collector current BC barrier ideality factor	2
VAF	forward Early voltage	300V
VAR	reverse Early voltage	100V
IK	knee current for dc high injection effect	0.1 A
BF	forward ideal current gain	500
BR	reverse ideal current gain	1000
ISE	saturation value for nonideal base current	2.7e-18 A
NE	ideality factor for nonideal forward base current	1.8
*ISEX	saturation value for emitter leakage diode	4e-24A
*NEX	ideality factor for emitter leakage diode	1.3
ISC	saturation value for intrinsic bc junction current	1.2e-14 A
NC	ideality factor for intrinsic bc junction current	2
*ISCX	saturation value for extrinsic bc junction current	5.2e-14 A
*NCX	ideality factor for extrinsic bc junction current	2
*FA	Factor for specification of avalanche voltage	0.995
*BVC	collector-base breakdown voltage BVcbo	28V
*NBC	exponent for BC multiplication factor vs voltage	6
*ICS	saturation value for collector-substrate current	1e-30 A
*NCS	ideality factor for collector-substrate current	2
RE	Emitter resistance	16 ohm
*REX	Extrinsic emitter leakage diode series resistance	2000 ohm
RBX	Extrinsic base resistance	55 ohm
RBI	Intrinsic base resistance	20 ohm
*RCX	Extrinsic collector resistance	10 ohm
*RCI	Intrinsic collector resistance	20 ohm
CJE	BE depletion capacitance at zero bias	14 fF
VJE	BE diode builtin potential for Cj estimation	1.384 V
MJE	Exponent for voltage variation of BE Cj	0.5
*CEMIN	Minimum BE capacitance	3 fF
FCE	Factor for start of high bias BE Cj approximation	0.975
*CJC	Intrinsic BC depletion capacitance at zero bias	8 fF
*VJC	Intrinsic BC diode builtin potential for Cj estimation	1.077 V
*MJC	Exponent for voltage variation of Intrinsic BC Cj	0.514
*CCMIN	Minimum value of intrinsic BC Cj	3fF
*FC	Factor for start of high bias BC Cj approximation	0.8
*CJCX	Extrinsic BC depletion capacitance at zero bias	7fF
*VJCX	Extrinsic BC diode builtin potential for Cj estimation	1.4V
*MJCX	Exponent for voltage variation of Extrinsic BC Cj	0.514
*CXMIN	Minimum extrinsic Cbc	4fF
XCJC	Factor for partitioning extrinsic BC Cj	1
*CJS	Collector-substrate depletion capacitance (0 bias)	0.05 fF
*VJS	CS diode builtin potential for Cj estimation	1.4V
*MJS	Exponent for voltage variation of CS Cj	0.01
TFB	Base transit time	0.3pS
*TBEXS	Excess BE heterojunction transit time	0.1pS
*TBCXS	Excess BC heterojunction transit time	0
*TFC0	Collector forward transit time	1pS
*ICRIT0	Critical current for intrinsic Cj variation	12mA
*ITC	Characteristic current for TFC	6mA
*ITC2	Characteristic current for TFC	30mA
*VTC	Characteristic voltage for TFC	10V
*TKRK	Forward transit time for Kirk effect	0.5pS
*VKRK	Characteristic Voltage for Kirk effect	10V
*IKRK	Characteristic current for Kirk effect	12mA
TR	Reverse charge storage time for intrinsic BC diode	350pS
TRX	Reverse charge storage time for extrinsic BC diode	350pS
*FEX	Factor to determine excess phase	0.25
*RTH	Thermal resistance from device to thermal ground	2200 C/W
*CTH	Thermal capacitance of device	3e-10 C/joule
KFN	BE flicker noise constant	0
AFN	BE flicker noise exponent for current	1.5
BFN	BE flicker noise exponent for frequency	1
XTI	Exponent for IS temperature dependence	2
XTB	Exponent for beta temperature dependence	-2.8
*TNE	Coefficient for NE temperature dependence	0
*TNC	Coefficient for NC temperature dependence	0
*TNEX	Coefficient for NEX temperature dependence	0
EG	Activation energy for IS temperature dependence	1.645
*EAA	Activation energy for ISA temperature dependence	-0.495V
*EAB	Activation energy for ISB temperature dependence	-0.1V
*EAE	Added activation energy for ISE temp dependence	0.105V
*EAB	Added activation energy for ISC temp dependence	0
*EAX	Added activation energy for ISEX temp dependence	0
*XRE	Exponent for RE temperature dependence	0.5
*XREX	Exponent for REX temperature dependence	0.5
*XRB	Exponent for RB temperature dependence	0.5
*XRC	Exponent for RC temperature dependence	0.5
*TVJE	Coefficient for VJE temperature dependence	-1.5e-3 V/C
*TVJCX	Coefficient for VJCX temperature dependence	-1.5e-3 V/C
*TVJCI	Coefficient for VJC temperature dependence	-1.5e-3 V/C
*TVJS	Coefficient for VJS temperature dependence	-1.5e-3 V/C
*XTITC	Exponent for ITC temperature dependence	0
*XTITC2	Exponent for ITC2 temperature dependence	0
*XTTF	Exponent for TF temperature dependence	0.75
*XTTKRK	Exponent for TKRK temperature dependence	0.6
*XTVKRK	Exponent for VKRK temperature dependence	0.6
*XTIKRK	Exponent for IKRK temperature dependence	0.6
*XRT	Exponent for RTH temperature dependence	1.2
*DTMAX	Maximum expected temperature rise above heatsink	1000 C

9.0 Goodies
Various files are available from our anonymous FTP site (sigma.ucsd.edu). Below is a brief description of the files.

README - Information on the files in the HBT_Modeling directory
hbestxls.zip - HBesT Ver 1.0 - HBT SPICE Parameter Estimator Spreadsheet (MS Excel Format)
hbestdoc.zip - HBest Ver 1.0 Documentation (MS Word Format)

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